Report 7 - Box-Cox Transformation (DxH)

if(!require(geoR)) install.packages("geoR")
if(!require(stats)) install.packages("stats")
if(!require(ggplot2)) install.packages("ggplot2")
if(!require(nortest)) install.packages("nortest")
if(!require(graphics)) install.packages("graphics")
if(!require(EnvStats)) install.packages("EnvStats")
if(!require(rcompanion)) install.packages("rcompanion")

Para realizar este estudo, utilizamos um dataset com 16.000 valores de entropia de permutação de Shannon e complexidade estatística referentes a ruídos brancos gerados artificialmente com amostras de tamanhos \(N \in \{10.000, 20.000, 30.000, 40.000, 50.000, 60.000, 70.000, 80.000, 90.000, 100.000\}\), aplicando todas as possíveis configurações de \(D \in \{3, 4, 5, 6\}\) e \(\tau \in \{1, 2, 3, 4\}\) aos descritores.

A disposição dos dados no plano \(HC\) pode ser observada abaixo:

HC.BP = data.frame("H" = numeric(16000), 
                   "C" = numeric(16000),
                   "Dist" = numeric(16000),
                   "D" = numeric(16000),
                   "N" = numeric(16000), 
                   stringsAsFactors=FALSE)
HC.BP$N = as.factor(rep(c(rep(1e+04, 100), rep(2e+04, 100), rep(3e+04, 100), rep(4e+04, 100), rep(5e+04, 100), rep(6e+04, 100), rep(7e+04, 100), rep(8e+04, 100), rep(9e+04, 100), rep(1e+05, 100)), 16))
file.csv = data.frame(read.csv("../Data/HC_series_fk0_16000.csv"))
HC.BP$H = file.csv[,1]
HC.BP$C = file.csv[,2]
HC.BP$Dist = HC.BP$C / HC.BP$H
HC.BP$D= as.factor(file.csv[,3])

Transformação de Box-Cox

dimension = c(3,4,5,6)
N = c(1e+04, 2e+04, 3e+04, 4e+04, 5e+04, 6e+04, 7e+04, 8e+04, 9e+04, 1e+05)
HC.test.3 = data.frame("H" = numeric(400), 
                       "Dist" = numeric(400),
                       stringsAsFactors=FALSE)
lm.3.box.cox = array(list(), 40)
b = cc = 0
for(i in 1:length(dimension)){
  for(j in 1:length(N)){
    cc = cc + 1
    a = c((((j - 1) * 100) + 1):(j * 100))
    elements = c(a + b, a + b + 1000, a + b + 2000, a + b + 3000)
    
    HC.test.3$H = HC.BP$H[elements]
    HC.test.3$Dist = HC.BP$Dist[elements]
    
    parameters = boxcoxfit(HC.test.3$Dist, lambda2 = F)
    lambda = parameters$lambda[1]
    if(lambda==0){T.norm = log(HC.test.3$Dist)}
    if(lambda!=0){T.norm = ((HC.test.3$Dist)^lambda - 1)/lambda}
    HC.test.3$Dist = T.norm
    
    lm.3.box.cox[[cc]] = lm(data = HC.test.3, formula = Dist ~ H)
  }
  b = b + 4000
}

HC.test.3$H = HC.BP$H[1:100]
HC.test.3$Dist = HC.BP$Dist[1:100]
parameters = boxcoxfit(HC.test.3$Dist, lambda2 = F)
lambda = parameters$lambda[1]
if(lambda==0){T.norm = log(HC.test.3$Dist)}
if(lambda!=0){T.norm = ((HC.test.3$Dist)^lambda - 1)/lambda}
HC.test.3$Dist = T.norm
plot(x = HC.test.3$H, y = HC.test.3$Dist)

ggplot(HC.test.3, aes(x = H, y = Dist)) +
  geom_point() +
  scale_fill_grey() +
  geom_line(aes(y = predict(lm.3.box.cox[[1]], HC.test.3, type = 'response'))) 

Analisando o plot do modelo:

plot(lm.3.box.cox[[1]])

hist(lm.3.box.cox[[1]]$residuals, breaks = 200)

Examinando o \(QQ-plot\) dos resíduos:

qqPlot(rstandard(lm.3.box.cox[[1]]))